Acceleration Due to Gravity (g)

When you drop a ball, it accelerates downward at approximately 9.8 m/s² — this is the acceleration due to gravity, universally denoted by the symbol g. Unlike a general variable, g has a specific physical meaning: it is the rate at which any freely falling object near Earth's surface gains speed every second, regardless of the object's mass. This remarkable fact, first demonstrated experimentally by Galileo Galilei in the late 16th century, was later explained theoretically by Newton's law of universal gravitation. The quantity g is intimately connected to the concept of gravitational field strength. The gravitational field strength at a point in space is defined as the gravitational force experienced per unit mass placed at that point. Near Earth's surface, this field strength has the value 9.8 N/kg, which is numerically identical to the free-fall acceleration of 9.8 m/s². This is not a coincidence — it is a direct consequence of Newton's second law (F = ma), which reveals that g simultaneously plays two roles: it is both an acceleration (m/s²) and a field strength (N/kg), two units that are dimensionally equivalent. Understanding g is foundational to almost every branch of classical mechanics. It governs the trajectory of projectiles, determines the weight of objects, drives the motion of pendulums, and shapes tidal forces across the solar system. While g ≈ 9.8 m/s² is the standard value used near Earth's surface, it is not a universal constant — it varies with altitude, latitude, and the mass and radius of the planet or moon in question. This article explores what g really means, how it is derived from first principles, and how gravitational field strength connects the abstract idea of a gravitational field to the measurable world.

Key takeaways

Newton's Law of Universal Gravitation
Applying Newton's Second Law
Defining g at Earth's Surface
g as Gravitational Field Strength
Weight and the Role of g

Newton's Law of Universal Gravitation

Newton's law states that the gravitational force between any two masses M and m separated by distance r is F = GMm/r². Here G is the universal gravitational constant (6.674 × 10⁻¹¹ N·m²/kg²), M is the mass of Earth (≈ 5.972 × 10²⁴ kg), m is the mass of any object near the surface, and r is the distance from Earth's centre to the object (approximately equal to Earth's radius R_E ≈ 6.371 × 10⁶ m at the surface).

Applying Newton's Second Law

By Newton's second law, the net force on a freely falling object of mass m equals its mass times its acceleration: F = ma. Setting the gravitational force equal to the net force gives GMm/r² = ma. Notice that the object's mass m appears on both sides of the equation. Dividing both sides by m cancels it completely, yielding: a = GM/r². This is the crucial step — it mathematically proves why all objects fall with the same acceleration regardless of their mass.

Defining g at Earth's Surface

At Earth's surface, r = R_E, so the free-fall acceleration becomes g = GM_E/R_E². Substituting values: g = (6.674 × 10⁻¹¹ × 5.972 × 10²⁴) / (6.371 × 10⁶)² = (3.986 × 10¹⁴) / (4.059 × 10¹³) ≈ 9.82 m/s². The small discrepancy from the textbook value of 9.8 m/s² arises because Earth is not a perfect sphere and rotates, effects we ignore in this idealisation.

g as Gravitational Field Strength

The gravitational field strength g at a point in space is defined as the gravitational force per unit mass: g = F/m (in vector form: g = F/m). It tells us the force that would act on each kilogram of mass placed at that point. At Earth's surface, g = F/m = (GMm/R_E²)/m = GM/R_E² — exactly the same expression as the free-fall acceleration. Therefore g = 9.8 N/kg = 9.8 m/s², confirming that gravitational field strength and free-fall acceleration are the same physical quantity expressed with equivalent units: 1 N/kg = 1 kg·m/s²/kg = 1 m/s².

Weight and the Role of g

Because the gravitational force on an object of mass m is F = mg, this force is called the object's weight W. Weight is not the same as mass — mass is an intrinsic property measured in kilograms, while weight is a force measured in newtons and depends on the local value of g. On the Moon, where g_Moon ≈ 1.62 m/s², a 60 kg person weighs only 60 × 1.62 = 97.2 N, compared to 60 × 9.8 = 588 N on Earth. Their mass, however, remains 60 kg everywhere.

The Key Equation

g = \frac{GM_E}{R_E^2}

g — Acceleration due to gravity (gravitational field strength) (m/s² or equivalently N/kg): The free-fall acceleration experienced by any object near Earth's surface; also equals the gravitational force per unit mass (field strength). Standard value: 9.8 m/s².
G — Universal gravitational constant (N·m²/kg²): A fundamental constant of nature with value 6.674 × 10⁻¹¹ N·m²/kg², appearing in Newton's law of universal gravitation.
M_E — Mass of Earth (kg): The total mass of Earth, approximately 5.972 × 10²⁴ kg. A larger planetary mass produces a stronger gravitational field and hence a larger g.
R_E — Radius of Earth (m): The distance from Earth's centre to its surface, approximately 6.371 × 10⁶ m. Because r appears squared in the denominator, even small increases in altitude noticeably reduce g.

Worked Example

Problem. A student stands on a planet whose mass is 3.20 × 10²³ kg and whose radius is 3.39 × 10⁶ m (these are the actual values for Mars). (a) Calculate the acceleration due to gravity on Mars's surface. (b) If the student has a mass of 72.0 kg, what is their weight on Mars? (c) What would their weight be on Earth (g_Earth = 9.80 m/s²)?

g_{\text{Mars}} = \frac{GM}{R^2} = \frac{(6.674\times10^{-11})(3.20\times10^{23})}{(3.39\times10^6)^2} \approx 3.73\,\text{m/s}^2

Answer. g_Mars ≈ 3.73 m/s²; weight on Mars ≈ 269 N; weight on Earth ≈ 706 N (mass = 72.0 kg on both).

Figure

Diagram

Origins · 1 minIsaac Newton and how this was discovered

Isaac Newton

1643–1727 · English

Newton formulated the law of universal gravitation in his landmark 1687 work Philosophiæ Naturalis Principia Mathematica. By showing that the same inverse-square force law governs both the fall of an apple and the orbit of the Moon, he unified terrestrial and celestial mechanics. His derivation that g = GM/R² is a direct consequence of this law, and his second law F = ma explained why all objects fall with the same acceleration regardless of mass — something Galileo had observed empirically but never fully explained mathematically. Newton used the known orbital period of the Moon (≈ 27.3 days) and its distance from Earth (≈ 60 Earth radii) to verify that the gravitational force at the Moon's distance was exactly 1/3600 of the surface value — perfectly consistent with his inverse-square law (since 60² = 3600). This 'Moon test' was his private proof that gravity extended into space with the right mathematical form.

Inverse-Square Law — The formula g = GM/r² is a direct expression of the inverse-square law — doubling the distance from Earth's centre reduces g by a factor of four. Understanding the inverse-square relationship is essential for predicting how g changes with altitude.
Gravitational Constant (G) — G is the proportionality constant that appears in the formula g = GM/R². Knowing G allows us to calculate g for any planet given its mass and radius, making it the bridge between the abstract law of gravitation and measurable quantities like g.
Gravitational Field Concept — g is the magnitude of the gravitational field vector at a point in space. The gravitational field concept explains why g can be treated as a property of space itself rather than requiring direct contact between masses.
Variation of g — While this article derives the standard surface value g ≈ 9.8 m/s², g is not constant — it varies with altitude (via r in GM/r²), latitude (Earth's equatorial bulge and rotation), and local geology. This concept explores all those variations quantitatively.
Free-Fall as 1D Motion — Once g is established as the constant downward acceleration of any freely falling object, the kinematic equations of 1D motion (v = u + gt, s = ut + ½gt²) can be applied directly to solve free-fall problems — g is the acceleration input for all such calculations.
Upward & Downward Motion Under Gravity — Whether an object is thrown upward or falls downward, g acts as a constant deceleration (going up) or acceleration (going down). Understanding that g is constant in both directions is key to analysing the full trajectory of vertically launched objects.
Distinction Between Mass & Weight — g is the link between mass (intrinsic, in kg) and weight (a force, in N): W = mg. Confusing mass and weight is a common error that dissolves once the role of g as a field strength — which varies by location — is clearly understood.
Weight on Earth, Moon & Space — Because g = GM/R² depends on a body's mass and radius, different planets and moons have different values of g, leading to different weights for the same mass. The worked example in this article (Mars) is a direct application of this idea.
Apparent Weight & Free-Fall — In free fall, every part of a body accelerates at g simultaneously, so there is no normal force and the apparent weight is zero — the sensation of weightlessness. This phenomenon is only understandable if one clearly separates the gravitational field strength g from the support forces that create the feeling of weight.
Gravitational Potential Energy Near Earth — The near-Earth formula for gravitational potential energy, E_p = mgh, uses g as the constant field strength approximation near the surface. The validity of this linear approximation — and when it breaks down — depends directly on understanding that g ≈ constant only when the change in altitude is much smaller than R_E.